Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I think that I am doing it right, but I am not sure. Out of 3 children there are 7 possibilities of having a girl out of 8 possible outcomes. I am dividing the 7 possibilities of having a girl by 3 children: 7/3=2.33. I am not real good at this, therefore, is the expected number of girls in a family of 3 the 2.3 I came up with or am I doing this completely wrong? Thank you for any help that I can get!

share|improve this question
2  
It cannot be right. If it were, the expected number of boys would also be $7/3$, giving an expected family size of $14/3$! –  André Nicolas Dec 3 '12 at 0:45

2 Answers 2

The family could have $0,1,2$, or $3$ girls. Of the $8$ possible sequences, only one gives them $0$ girls, so the probability of $0$ girls is $\frac18$. There are $3$ ways for them to have $1$ girl, however: GBB, BGB, and BBG. Thus, the probability of $1$ girl is $\frac38$. You can check similarly that they have $2$ girls with probability $\frac38$ and $3$ girls with probability $\frac18$.

The expected number of girls is a weighted average of the possible numbers, where the weight of a given number is its probability of occurring. Thus, the expected number of girls is

$$\frac18\cdot0+\frac38\cdot1+\frac38\cdot2+\frac18\cdot3=\frac{12}8=1.5\;.$$

You can also think of this as the average over all of the $8$ equally likely birth orders of three children: add up the number of girls in BBB, BBG, BGB, GBB, BGG, GBG, GGB, and GGG, and divide by the $8$ possibilities to get (again) $\frac{12}8=1.5$.

Finally, you can observe that since boys and girls are equally likely, the expected number of boys should be the same as the expected number of girls. Since each family has $3$ children altogether, and we expect half of them on average to be girls, we expect on average $\frac32=1.5$ girls per family.

share|improve this answer

Each child has a $\frac12$ probability of being a girl. This is the expected number of girls among each one of the three children. Expectations are additive, so the expected number of girls among all three children is $\frac12+\frac12+\frac12 = \frac32 = 1.5$.

Similarly, the expected number of spots on one die is 3.5, so the expected number of spots on two dice added together is 3.5 + 3.5 = 7.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.